Codes for error detection, good or not good

TL;DR

This paper investigates the effectiveness of linear and non-linear codes for error detection on q-ary symmetric channels, establishing bounds on code length for good error detection properties.

Contribution

It introduces a threshold function mma(d,k) indicating when codes and their duals cease to be effective for error detection, with approximations for extreme parameter values.

Findings

Existence of a length threshold mma(d,k) for effective error detection

Approximate bounds for mma(d,k) when d bb k or k bb d

Extension of results to non-linear codes

Abstract

Linear codes for error detection on a q-ary symmetric channel are studied. It is shown that for given dimension k and minimum distance d, there exists a value \mu(d,k) such that if C is a code of length n >= \mu(d,k), then neither C nor its dual are good for error detection. For d >> k or k << d good approximations for \mu(d,k) are given. A generalization to non-linear codes is also given.